Jiansheng Geng

A KAM theorem for Hamiltonian networks with long ranged couplings

We consider Hamiltonian networks of long-ranged and weakly coupled oscillators with variable frequencies. By deriving an abstract infinite dimensional KAM type of theorem, we show that for any given positive integer N and a fixed, positive measure set of N variable frequencies, there is a subset of positive measure such that each corresponds to a small amplitude, quasi-periodic breather (i.e. a solution which is quasi-periodic in time and exponentially localized in space) of the Hamiltonian network with N-frequencies which are slightly deformed from ω.

Quasi-periodic Solutions for One-Dimensional Nonlinear Lattice Schrödinger Equation with Tangent Potential

In this paper, we construct time quasi-periodic solutions for the nonlinear lattice Schrodinger equation

A KAM Theorem for Hamiltonian Partial Differential Equations in Higher Dimensional Spaces

{\rm i}\dot{q}_n+\epsilon (q_{n+1}+q_{n-1}) +\tan\pi(n\tilde{\alpha}+x)q_n+\epsilon|q_n|^2q_n=0

A KAM theorem for one dimensional Schrödinger equation with periodic boundary conditions

,

Quasi-Periodic Solutions in a Nonlinear Schrödinger Equation

n\in\mathbb{Z},

An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation

where

KAM tori for higher dimensional beam equations with constant potentials

\tilde{\alpha}

KAM tori of Hamiltonian perturbations of 1D linear beam equations

satisfies a certain Diophantine condition and

Almost periodic solutions for a class of higher-dimensional beam equations

x\in\mathbb{R}/\mathbb{Z}

Invariant tori of full dimension for a nonlinear Schrödinger equation

. We prove that for